Self-converse Mendelsohn designs with block size 6q
Abstract
A Mendelsohn design MD(v, k, λ) is a pair (X, B) where X is a v-set together with a collection B of cyclic k-tuples from X such that each ordered pair from X is contained in exactly λ cyclic k-tuples of B. An MD(v, k, λ) is said to be self-converse, denoted by SCMD(v, k, λ) = (X, B, f), if there is an isomorphic mapping f from (X, B) to (X, B-1), where B-1 = {B-1 = 〈xk, xk-1 , . . . , x2, x1〉 : B = 〈x1 , . . . , xk〉 ∈ B}. The existence of SCMD(v, 3, λ), SCMD(v, 4, 1), SCMD(v, 5, 1) and SCMD(v, 4t + 2, 1) has been completely settled, where 2t + 1 is a prime power. In this paper, we investigate the existence of SCMD(v, 6q, 1), where gcd(q, 6) = 1. In particular, when q is a prime power, the existence spectrum of SCMD(v, 6q, 1) are solved, except possibly for two small subclasses. As well, our conclusion extends the existence results for MD(v, k, 1) also.
